Method for Constructing a Commutative Algebra of Hypercomplex Numbers

Until now, it was believed that, unlike real and complex numbers, the construction of a commutative algebra of quaternions or octonions with division over the field of real numbers is impossible in principle. No one questioned the existing theoretical assertion that quaternions, octonions, and other hypercomplex numbers cannot have the commutativity property. This article demonstrates the following for the first time: (1) the possibility of constructing a normed commutative algebra of quaternions and octonions with division over the field of real numbers; (2) the possibility of constructing a normed commutative algebra of six-dimensional and ten-dimensional hypercomplex numbers with division over the field of real numbers; (3) a method for constructing a normed commutative algebra of N-dimensional hypercomplex numbers with division over the field of real numbers for even values of N; and (4) the possibility of constructing a normed commutative algebra of other N-dimensional hypercomplex numbers with division over the field of real numbers. The article also shows that when using specific forms of representation of unit vectors, the product of vectors has the property of commutativity. Normed commutative algebras of N-dimensional hypercomplex numbers can be widely used to solve many topical scientific problems in the field of theoretical physics for modeling force fields with various types of symmetry, in cryptography for developing a number of new cryptographic programs using hypercomplex number algebras with different values of dimension, and in many other areas of fundamental and applied sciences

Synthesis, Properties and Spatial Structure of 4-[(3,5-Dimethyl-1,2-oxazol-4-yl)sulfonyl]cytisine

This article has studied the synthesis of a new derivative of the known alkaloid cytisine contained in the seeds of plants of Cytisus laburnum L. and Thermopsis lanceolata R.Br., both of the Lugiminosae family. The new compound has been obtained from two biologically active compounds, such as isoxazole and cytisine. It has been demonstrated that the reaction led to the single-stage method under very mild conditions to obtain 4-[(3,5-dimethyl-1,2-oxazol-4-yl)sulfonyl]cytisine. This class of compounds is promising for obtaining the new biologically active compounds. This article has examined, in detail, a structure with using the 1H and 13C NMR and two-dimensional NMR spectroscopy of COSY (1H-1H), HMQC (1H-13C) and HMBC (1H-13C). As a result, the homo- and heteronuclear spin-spin couplings should be established. The X-ray diffraction analysis has determined the spatial structure of a new derivative based on the cytisine alkaloid. Thus, its hemorheological activity has been studied